Topological manipulations on $\textit{ℝ}$ symmetries of Abelian gauge theory

Performing topological manipulations is a fruitful way to understand global aspects of Quantum Field Theory (QFT). Such modifications are typically controlled by the notion of Topological QFT (TQFT) coupling across different codimensions. Motivated by the recent developments involving non-compact TQFTs as the Symmetry Topological Field Theory (SymTFT) for continuous symmetries, we realize topological manipulations on global $\textit{ℝ}$ symmetries via TQFT coupling in the simple context of non-compact abelian gauge theory. Namely, by inserting the background fields for $\textit{ℝ}$ symmetries into non-compact TQFTs on spacetime, we study the topological gaugings. Furthermore, we explore topological defects in non-compact theories by employing the said manipulations on the half-space, which are analogous to duality defects in compact gauge theories. We examine the action of these defects on the local and extended operators, and discuss their algebra. As opposed to the duality defects, our defects act invertibly on the spectrum. To further understand their role, we discuss the way they mix with other topological defects and the resulting global symmetries. We also comment on possible applications of these ideas to the bosonic String Theory. After studying defects, we provide a detailed inspection of manipulations on $\textit{ℝ}$ and $\textit{ℝ}$ symmetries in abelian gauge theory. Notably, we develop a novel deformation tailored for flat gauging subgroups of $\textit{ℝ}$ symmetries, and provide an extension of it for $\textit{ℝ}$ symmetries. We show that this new simple deformation captures all the topological boundary conditions of the corresponding non-compact SymTFTs as the deformation parameter is varied, which the ordinary non-compact BF coupling cannot do.